Improved energy harvesting from low-frequency small vibrations through a monostable piezoelectric energy harvester

Mechanical Systems and Signal Processing 117 (2019) 594–608

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Improved energy harvesting from low-frequency small vibrations through a monostable piezoelectric energy harvester Kangqi Fan a,⇑, Qinxue Tan a, Haiyan Liu b, Yiwei Zhang a, Meiling Cai a a b

School of Mechano-Electronic Engineering, Xidian University, Xi’an 710071, China Department of Rehabilitation Medicine, Shaanxi Second Provincial People’s Hospital, Xi’an 710005, China

a r t i c l e

i n f o

Article history: Received 6 May 2018 Received in revised form 16 July 2018 Accepted 1 August 2018

Keywords: Piezoelectric effect Energy harvesting Monostable configuration Low-frequency vibration Random excitation

a b s t r a c t Scavenging energy from low-frequency and low-level excitations has always been a huge challenge for the piezoelectric energy harvesting since the frequencies of ambient excitations are usually below the device’s operating frequency and small excitations may fail to actuate the device to produce usable electricity. To remedy this key issue, a piezoelectric energy harvester with stoppers (PEHS) has been proposed by the authors. The stoppers and the magnetically attractive coupling employed in the PEHS make the device monostable, removing the requirement for overcoming the potential barrier that normally appears in a bistable or tristable system. A theoretical model for the PEHS is established and experimentally validated, with which the PEHS is investigated under both harmonic excitations and random excitations. The results indicate that the operating frequency range of the PEHS can be tuned toward the lower frequency by changing the (mass-magnet) gap between the tip mass and the external magnets, making the efficient energy harvesting from low-frequency excitations possible. For a given harmonic excitation, the PEHS can provide a larger power output and wider operating bandwidth than the linear PEH no matter what way the frequency sweep is conducted. Moreover, compared with the linear PEH, improved power output can also be attained under the Gaussian white noise with a small intensity, enabling the PEHS to deliver useful power even in the presence of small random excitations. Although the optimal PEHS configuration in terms of mass-magnet gap is found to vary slightly with the excitation levels, there exists a certain gap that can guarantee the optimal or near optimal performance of the PEHS under both low-level harmonic excitations and low-intensity random excitations, demonstrating the harvester’s superior adaptation to the ambient excitations with variable strengths. Ó 2018 Elsevier Ltd. All rights reserved.

1. Introduction Harvesting ambient vibration energy to generate electricity has been increasingly considered as a pivotal point for the development of self-sustained micro-powered electronic devices [1–3]. The almost omnipresent vibration energy can be captured to recharge electrochemical batteries, currently the main power source of micro-powered devices, or even to replace batteries to directly power these devices, contributing to the realization of completely self-sufficient microsystems [4,5]. In accordance with human society’s call for renewable green energy, vibration energy harvesting technologies

⇑ Corresponding author. E-mail address: [email protected] (K. Fan). https://doi.org/10.1016/j.ymssp.2018.08.001 0888-3270/Ó 2018 Elsevier Ltd. All rights reserved.

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exploiting the ambient waste energy can provide a sustainable power source and then demonstrate potential applications in a variety of fields, such as embedded sensors/microsystems in buildings and structures, wearable devices, medical implants, and wireless sensor networks [6–11]. Among various vibration energy harvesting mechanisms, the piezoelectric energy harvesting has attracted most attention due to its simple structure and high energy density [9,12,13]. Conventional piezoelectric energy harvesters (PEHs) are generally designed as a linear resonator and then operate effectively only in a limited frequency range around the resonance, leading to poor harvesting efficiency under broadband ambient vibrations [8,14–16]. One straightforward approach for solving this key issue is integrating multiple linear harvesters with different but close resonant frequencies in one device to construct an array harvester [17–19]. It is not surprising that the array harvester can cover a wide frequency range but at the cost of reducing the power density. A retrofitted version of array harvesters is the multi-modal energy harvester [20], which is realized by attaching an additional oscillator or dynamic magnifier to the linear PEH so that two close resonances can be achieved in one structure [4,8,21–24]. Another strategy for broadband energy harvesting is the tuning mechanism, which shifts the natural frequency of a harvester into the spectral region where most of the vibration energy is available [25]. Although the tuning mechanism can be implemented via several different ways, such as applying axial load on a cantilever beam [26] and changing the position of the proof mass [17], the increased complexity in design and extra power consumption for active tuning may overwhelm the performance improvement brought about by this strategy. In recent years, nonlinearities generated by the magnetic interaction have been widely exploited to improve the harvester performance as the nonlinearity-induced bending of response curves can be employed to cover a wider frequency range [27]. A typical nonlinear PEH is composed of a piezoelectric cantilever beam with one magnet affixed at the free end and the other one at the enclosure of the device [14,28–32]. For small spacing of the two magnets with repulsive arrangement, this nonlinear structure exhibits bistable behavior [33], whereas the large spacing or attractive arrangement makes the PEH monostable. A comprehensive study by Tang et al. [29] indicates that the bistable PEH works very well when the spacing of the two repulsive magnets is set close to the monostable-to-bistable transition region. Another commonly adopted approach for constructing nonlinear PEHs is based on the Duffing oscillator that consists of a ferromagnetic cantilever beam with two magnets positioned symmetrically near the free end [15]. By bonding two piezoelectric patches to the root of the cantilever beam, the Duffing oscillator based piezomagnetoelastic structure is capable of extracting energy from broadband vibrations [34]. This piezomagnetoelastic structure was further ameliorated by Cao et al. [35] via attaching a magnet at the beam’s free end and making the two external magnets rotatable. Depending on the angular orientation of the two external magnets, the improved piezomagnetoelastic structure can be monostable, bistable, or even tristable [36,37]. On the other hand, if the magnet at the beam’s free end is arranged to be repelled by the two external magnets, both softening and hardening responses can be attained through moving the two external magnets along the beam’s longitudinal direction [38]. Moreover, the magnetic interaction has also been explored to tune the PEH’s resonance frequency [39], stimulate the internal resonance [8,40], and enable the energy exchange between different vibration modes [4,41]. Furthermore, special structural benefits have been exploited in recent years to expand the working bandwidth of an energy harvesting device. For example, Wei and Jing [42] proposed a novel nonlinear energy harvesting system that consists of a lever system and an X-shape supporting structure to achieve tunable resonant frequency and large energy harvesting bandwidth. In addition, diverse harvesters/nanogenerators have also been developed based on the exploitation of novel materials, such as the porous polymer composite membrane based nanogenerator [43], fish-skin-based nanogenerator [44], prawn shells made nanogenerator [45], fish gelatin nanofibers based harvester [46], and the harvester made of modified polyvinylidene fluoride (PVDF) that can convert both mechanical and thermal energies into electrical power [47]. Although various broadband strategies have enhanced the PEH’s robustness to the varying excitation frequencies and then its performance, the enlarged operating bandwidth may still fail to cover the excitation sources of low frequencies, such as vehicle motion, machine vibration, and wind-induced vibration, which usually occur at comparatively low frequencies [29,48–50]. This weakness is usually tackled by what is called the frequency up-conversion technique that generally involves two oscillators: a low-frequency oscillator, which is designed to match the excitation frequency better; a high-frequency oscillator that absorbs energy from the low-frequency oscillator and generates electricity. The transfer of mechanical energy from the low-frequency oscillator to the high-frequency oscillator is mainly performed by mechanical impact and magnetic coupling. Since the energy loss caused by the mechanical impact is normally unavoidable, magnetic coupling has drawn increasing attention in recent years with intent to minimize the energy dissipated in the frequency up-converting PEHs [48,50–52]. For the aforementioned nonlinear PEHs, striking improvement in the operating bandwidth has been demonstrated by several studies provided that the excitation is sufficiently large and/or the appropriate frequency sweep operation (upward or downward sweep) can be guaranteed. Otherwise, the above nonlinear PEHs show no obvious superiority to the conventional PEH. To address this issue, we [53] proposed a new mechanism of utilizing the monostable PEH with stoppers (PEHS) to achieve wide operating bandwidth under low-level excitations. The PEHS consists of a piezoelectric beam with a magnetic tip mass, two external magnets, and two stoppers. The two external magnets are symmetrically placed near the beam’s free end so that attractive force is applied on both sides of the beam. The two stoppers are employed to confine the beam’s deflection range within which the elastic force dominates the magnetic force, making the device monostable. A preliminary analysis has shown that the PEHS can outperform the linear PEH in terms of operating bandwidth and peak voltage under both upward and downward frequency sweeps [53]. However, this observation is based only on the harmonic excitation with one

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constant magnitude. It is not clear whether the PEHS is superior to the linear PEH under various excitation levels, especially under the random excitations, which are close to the practical scenarios in the environment. This work presents extensive investigations on the PEHS to reveal its performance under not only harmonic excitations but also random excitations. First, a theoretical model for the PEHS is established and experimentally validated. With the model, voltage and power responses of the PEHS under sinusoidal excitations of various magnitudes are determined, and then effects of varying the positions of external magnets and stoppers on the voltage and power outputs of the PEHS are revealed. Subsequently, a series of simulations are performed to examine the performance of the PEHS under random vibrations with Gaussian distribution. Comparison between the PEHS and the linear PEH is made to demonstrate the improved performance achieved by the former. The theoretical and experimental studies show that the attractive arrangement of magnets in the proposed PEH makes the response curve of the PEH shift to the lower frequency, which is conducive to the energy harvesting from low-frequency excitations. Moreover, the operating bandwidth of the PEH can be tuned via varying positions of the external magnets. In addition, the monostable characteristic of the proposed PEH removes the need for overcoming the potential barrier that normally occurs in multi-stable PEHs, enabling the PEH to generate high power output under small harmonic excitations or random excitations.

2. PEHS configuration and modeling 2.1. PEHS architecture The PEHS is composed of a piezoelectric (bimorph) cantilever beam with two magnets fixed at the free end, two stoppers, and two external magnets fixed to the enclosure of the device, as shown in Fig. 1. The two external magnets are symmetrically placed near the free end of the beam, and the magnets used in the PEHS are arranged in such a way that attractive force is applied on both sides of the beam. For such a configuration, the magnetic tip mass and one of the two external magnets would snap together if their initial gap (D) is small, whereas a large value of D could markedly weaken the magnetic coupling between the beam and external magnets and then lower the performance enhancement achieved by the magnetic interaction induced nonlinearity. To maintain a moderate degree of magnetic coupling and in the meanwhile avoid locking up the beam by the external magnets, two stoppers are employed here to confine the beam’s deflection in a range within which the elastic force is always larger than the magnetic force, which makes the system monostable. For ease of presentation, from here onward in this paper, the initial gap D between the magnetic tip mass and external magnets is termed as ‘mass-magnet gap’. The PEHS differs from the monostable PEH proposed by Stanton et al. [38] in that the attractive interaction instead of repulsive interaction is exerted on both sides of the beam in the PEHS. The PEHS is also different from the tunable PEH reported by Challa et al. [39]; in the latter the attractive force and repulsive force are applied respectively on the beam’s two sides to adjust the resonance frequency of the device but no effort was made for scavenging energy from small excitations. A similar harvester with bistability, which was first developed by Erturk et al. [15] and had been retrofitted by Cao et al. [35–37], is realized based on what is now called the Duffing oscillator. The key difference between the PEHS and the Duffing oscillator-based harvester is that the former exhibits the typical softening response and the latter responds to the applied excitation in a hardening way. The softening response, as will be shown in this study, could shift the operating frequency band of the harvester toward the left (i.e., lower frequency), which enables the harvester to capture energy from lower-frequency excitations. Moreover, the attractive configuration between the magnetic tip mass and the two external magnets could increase the deflection of the beam and then improve the voltage and power outputs, enabling the harvester to deliver usable electrical output under small excitations. On the contrary, for the Duffing oscillator-based harvester, the frequency response curve is bended toward the right and large excitations are required to overcome the potential barrier

Fig. 1. Schematic diagram of the proposed PEHS with the beam’s axis perpendicular to the ground.

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so that the beam can oscillate between two stable equilibrium positions (or potential wells). Under small excitations, the vibration of the beam is almost confined within one potential well or occurs in the chaotic form, making the nonlinear harvester fail to outperform its linear counterpart [34]. Scavenging energy from low-frequency excitations with small magnitudes has always been an enormous challenge since it is usually difficult to build a typical cantilevered (linear) harvester with sufficiently low natural frequency. This problem is currently addressed by the frequency up-conversion technique, in which both low-frequency and high-frequency oscillators are required. Shortcomings of the frequency up-conversion technique include complicated design and reduced power density. For the proposed PEHS, the operating frequency range can be shifted to the lower frequency by simply tuning the massmagnet gap, which adds little complexity to the harvester’s structure, making the energy extracting from low-frequency excitations more feasible. Moreover, compared with harvesters reported in the literature, the proposed PEHS can generate high voltage and power outputs under both upward and downward frequency sweeps, yielding a large working bandwidth. Furthermore, the PEHS is particularly suited for capturing energy from low-level excitations since there is no need for overcoming the high potential barrier that normally appears in a multi-stable PEH. This design can also been extended to electromagnetic and electrostatic energy harvesting with similar structures. 2.2. Modeling Accurate modeling of a PEH is required to begin with the Euler-Bernoulli equation with appropriate boundary conditions [54]. However, since the dynamic response around the first mode is the primary concern in energy harvesting, governing equations of the PEHS under the x-directional excitation a can be simplified as

(

m€x þ gx_ þ kx þ HV þ f m ¼ ma Hx_ þ C P V_ ¼ I;

ð1Þ

where m is the equivalent mass; g is the damping coefficient; k is the equivalent stiffness; H is the equivalent electromechanical coupling coefficient; CP is the clamped capacitance of the piezoelectric layer; V and I are the voltage and current outputs from the piezoelectric beam, respectively; fm = f01 (D  x) + f02 (D + x) is the magnetic force acting on the magnetic mass with subscripts 0, 1, and 2 representing the magnetic mass, left magnet, and right magnet. It is worth mentioning that x denotes the displacement of the free end of the beam relative to the frame of the harvester, and a ¼ € x0 with x0 representing the absolute motion of the frame along the x direction. The magnetic coupling in the PEHS can be introduced with permanent magnets of various geometries. For two cylindrical magnets i and j, the magnetic force fij between them can be calculated by [55,56]

" # 1 1 1 1 1 pl0 r4 M2 2   þ  2  2 ; 4 d ðd þ t i Þ2 d þ tj d þ ti þ tj

f ij ðdÞ ¼ 

ð2Þ

where l0 is the vacuum permeability; r is the radius; M is the magnetization; ti and tj are the thicknesses of magnets i and j, respectively; d is the gap between them. For magnets 0 and 1 in the PEHS, d = D + x, whereas d = D  x for magnets 0 and 2. The potential energy for the piezomagnetoelastic system can be determined by integrating over the elastic force and magnetic force, i.e.,

Z

Z

x

EðxÞ ¼

kx dx  0

x

f m dx ¼ D

1 2 kx þ E01 þ E02 ; 2

ð3Þ

where E01 is the magnetic potential energy between the beam and the left magnet, and E02 the potential energy between the beam and the right magnet. For the beam and one of the two external magnets, the potential energy is

Z

Eij ðdÞ ¼ 

x

f ij ðdÞdx ¼ D

  1 1 1 1 1 þ C; pl0 M2 r4 þ   4 d þ ti d þ tj d d þ ti þ tj

ð4Þ

where C is the integration constant, indicating the magnetic potential energy when the magnetic tip mass is at its initial equilibrium position (x = 0), i.e.,

C¼

  1 1 1 1 1 : pl0 M2 r4 þ   4 D D þ ti þ tj D þ ti D þ tj

ð5Þ

To solve the governing equations, Eq. (1) is expressed in the state-space form as

8 9 8 9 u > < u_ 1 > < maf Hu2ku gu > = > = 3 1 2 m u_ 2 ¼ m > > > > :_ ; : ; H u2 u3  RC u3 Cp p

ð6Þ

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_ and u3 = V. The equations given by Eq. (6) can be solved using the finite difference where the state variables are u1 = x, u2 = x, _ and V. In this study, the MATLAB software is used to model the monostable PEH and its linear counterpart, method for x, x, and the finite difference method is programmed to obtain the output responses of the two harvesters. To illustrate the design of the proposed PEHS, the force–displacement relationship and potential energy for the piezomagnetoelastic system are depicted in Fig. 2 for different values of D with device parameters given in Table 1. It should be noted that the direction of the elastic force in Fig. 2 is reversed for showing a better comparison between the elastic force and the magnetic force. It is obvious that, for an overly small mass-magnet gap (e.g., D = 12 mm), the magnetic force dominates over the elastic force, causing the magnetic mass and the external magnet snap together and thus disabling the harvester if the magnetic mass deviates from its initial (unstable) equilibrium position (x = 0) slightly. On the contrary, a sufficiently large D (e.g., D = 22 mm) can make the elastic force larger than the magnetic force (within the maximum allowable deflection of the beam), but the weak magnetic coupling induced by the large value of D fails to bring about noticeable improvement in the energy harvesting performance. To produce large beam deflection and then high voltage output under low-level excitations, the ideal configuration is expected to provide a comparatively strong magnetic coupling between the beam and external magnets and in the meanwhile make the system monostable. This desirable configuration can be achieved using a mediate value of D (e.g., D = 16 mm), which results in one stable equilibrium position at x = 0 and two unstable equilibrium positions at x = ±xc, as shown in Fig. 2 (a). However, if the beam deflects beyond xc (or xc), the magnetic mass and magnet 2 (or 1) would also snap together, making the device unworkable. To solve this problem and keep the system monostable, two stoppers are employed to confine the deflection of the beam within the range [xc, +xc], as shown in Fig. 1. A side effect of using stoppers is the contact or collision between the magnetic mass and external magnets, which may be induced by overly large excitations. The effect of the contact or collision on the dynamic response of the system can be described by the change in the velocity of the magnetic mass before (v) and after (v+) the contact or collision, i.e., v+ =  ev with e = 0.3 indicating the energetic coefficient of restitution [13,57]. It is worth noting that the perfect positioning of the two stoppers at ±xc is scarcely realized in practice. However, a small deviation from the unstable equilibrium points, particularly, when the stoppers are biased toward the stable equilibrium position (x = 0), does not alter the PEHS dynamic response significantly. For the device parameters given in Table 1, the variation of the unstable equilibrium position xc with D is plotted in Fig. 3, which exhibits an approximate linear correlation between xc and D.

3. Experimental validation and numerical simulation According to the analyses given in Section 2, we can see that the PEHS response is subject to the strength of magnetic coupling between the beam and external magnets, which is dependent on the mass-magnet gap D. To obtain a reasonable prediction of the PEHS output, a set of experiments has been designed to validate the theoretical model and reveal the PEHS voltage outputs under both upward and downward frequency sweeps. The PEHS dynamic response and electrical outputs are then fully investigated through numerical simulations.

A

A

-xC

xC

(a)

(b)

Fig. 2. (a) Elastic force and magnetic force and (b) potential energy for various values of D.

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K. Fan et al. / Mechanical Systems and Signal Processing 117 (2019) 594–608 Table 1 System parameters. Parameters

Values

m (g) k (N m1) g (N s m1) H (N V1) CP (nF) M (A/m) r (mm) t0 (mm) t1 (mm) t1 (mm)

1.6 36.8 0.012 0.00012 14.7 1.05  106 3 6 3 3

Fig. 3. Unstable equilibrium position xc versus mass-magnet gap D.

3.1. Experimental validation The setup used for experimental validation is shown in Fig. 4 along with a close-up view of the PEHS configuration. The piezoelectric cantilever beam is constructed from a brass substrate sandwiched between two piezoelectric patches (PZT-5H, Konghong New Material Corp., model: Single crystal chip) that are electrically connected in series. The brass substrate has a dimension of 63  7  0.1 mm3, whereas the size of the piezoelectric patch is 30  7  0.25 mm3. Among the four cylindrical magnets (Nd-Fe-B), two of them are attached at the beam’s free end and the other two (external magnets) are fixed at the duralumin frame. A set of customized fixture made of duralumin is employed to facilitate shifting positions of the external magnets. Similarly, another set of fixture is devised to adjust positions of the stoppers that are machined from polymethyl

Vibration controller

Computer

Power amplifier

Fixture

Piezoelectric beam

Accelerometer Energy Vibration exciter harvester

Magnets

Oscilloscope Power amplifier

Accelerometer

Vibration controller

Oscilloscope

Harvester & shaker

(a)

Stoppers External magnets

(b) Fig. 4. (a) Experimental setup and (b) PEHS prototype.

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methacrylate. All bolts (excluding the longest) and nuts are made of brass to alleviate the possibility of magnetic field interference. Moreover, the PEH is fixed on the fixture with the axis of the piezoelectric beam perpendicular to the ground so as to eliminate the mass effect of the magnets on the dynamic response. If the two external magnets are removed, the system will degrade to a conventional linear PEH. In the experiment, the harvester is attached to the shaft of a vibration exciter (HEV-500, Nanjing Foneng Corp.) through a customized clamp made of duralumin to distance the harvester away from the exciter. A long distance between the harvester and the exciter could make the magnetic field interference from the exciter ignorable. The exciter is driven by a vibration controller (VT-9008-4, ECON) through a power amplifier (HEV-500G, Nanjing Foneng Corp.) to provide the base excitation that is measured using an accelerometer (CXL04GP3, Crossbow). The harvester response under various conditions is displayed and recorded by an oscilloscope (Rigol DS1074Z-S). System parameters of the fabricated prototype are given in Table 1, which are determined from the experimental rig using a model identification procedure presented by Tang and Yang [4]. First, we perform frequency sweep for the PEHS (D = 19 mm, d = xc = 10.2 mm) and its linear counterpart using a sinusoidal excitation with an amplitude of 0.5 g (g = 9.8 m/s2). In this case, the contact or collision between the magnetic mass and external magnets does not happen, and then the data obtained can reveal the PEHS output under normal operating conditions. Results regarding the frequency response of the voltage output under various settings are shown in Fig. 5, from which four important features can be identified: (1) the PEHS exhibits typical softening response; (2) response curve of the PEHS bends toward the lower frequency; (3) wide operating bandwidth is achieved by the PEHS compared with the linear PEH; (4) theoretical peak voltage outputs of the PEHS under both upward (40.9 V at 19.8 Hz) and downward (44.1 V at 18.9 Hz) sweeps are larger than that (34.2 V at 24.2 Hz) of the linear PEH. It is evident that the agreement between the model and the experimental data is good although the peak voltage outputs obtained by simulation are slightly larger than those observed by experiment (30.4 V from linear PEH, 32 V under upward sweep, and 33.4 V under downward sweep). The discrepancy in the magnitude of the voltage output is probably due to the change of damping in the experiment with the excitation frequency and the magnetic force introduced, and the rotation at the beam’s free end, which are not taken into account in the simulations. To reveal the PEHS voltage response in the case that contact or collision between the magnetic mass and external magnets occurs, frequency sweep for the PEHS with a smaller value of D (16 mm) is conducted using a sinusoidal excitation with an amplitude of 0.4 g. For comparison, the voltage output generated by the linear counterpart is also pictured, as shown in Fig. 6. It is apparent that the four features shown above are still observable in this case. However, due to the presence of the stoppers, the maximum deflection of the beam is confined, leading to frequent contact or collision between the magnetic mass and external magnets and yielding a plateau in the voltage response curve of the PEHS. The simulations show that the peak voltage corresponding to the plateau in the response curve is approximately 33.2 V, whereas the peak voltage measured by experiment is around 30.1 V. This discrepancy may result from the imperfect positioning of the stoppers and the difference of damping in the experiment and simulation. In addition, a constant energetic coefficient of restitution e used in the simulation may fail to precisely describe the energy loss induced by the collision between the magnetic mass and external magnets, which may be different at various excitation frequencies. Despite the small differences, simulations predict the same trends of nonlinear behavior as experiments. As show above, if the mass-magnet gap D is large, the coupling between the beam and external magnets is weak, and then a small reduction in the magnetic interaction between them induced by the rotation of the magnet affixed at the free end of the beam is negligible. Although the coupling between the beam and external magnets is strengthened with decreasing the value of D, two stoppers are employed to confine the deflection range of the beam and make the harvester monostable. For the adopted values of D and the excitation levels used in this study, the maximum deflection of the beam is <5 mm, which is approximately 7.1% of the beam length. Therefore, the change in the magnetization due to the rotation of the beam’s free end cannot change the dynamic response significantly. That is why a constant magnetization is generally utilized in the reported studies [4,38,39].

3.2. Output response under harmonic excitations With the validated model, further simulations are performed to reveal the output response of the proposed PEHS under sinusoidal excitations with various amplitudes, as shown in Figs. 7-9. It is obvious that, under both upward and downward sweeps with a sweep rate of 0.03 Hz/s, the PEHS can outperform the linear PEH in terms of voltage output and operating bandwidth. For the PEHS, decreasing the mass-magnet gap D could strengthen the magnetic coupling between the beam and external magnets, resulting in larger voltage output and more obvious softening response. Moreover, the shift of the operating frequency range toward the lower frequency may be achieved by increasing the magnetic coupling, i.e., reducing the value of D. This feature is useful in designing very low-frequency energy harvesters where it is difficult to build a linear resonator of sufficiently low frequency. Considering that the attractive coupling between the beam and external magnets contributes to the increase of the voltage output, the PEHS is also well suited for scavenging energy from low-level excitations since it does not have to overcome the potential barrier that normally appears in bistable or tristable harvesters. However, an overly strong magnetic coupling or a too small value of D may pose an adverse effect on the PEHS voltage output because the beam is confined to deflect within a smaller range (i.e., smaller value of d) to keep the harvester monostable.

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(a)

(b) Fig. 5. Open-circuit voltage output from PEHS and its linear counterpart (D = 19 mm, d = xc = 10.2 mm, and a = 0.5 g): (a) simulation and (b) experiment.

(a)

(b) Fig. 6. Open-circuit voltage output from PEHS and its linear counterpart (D = 16 mm, d = 4 mm, xc = 5.5 mm, a = 0.4 g): (a) simulation and (b) experiment.

For example, frequent contact or collision between the magnetic mass and external magnets occurs for D = 15 mm (d = xc = 3.2 mm) and a = 0.35 g, which limits the maximum voltage output. A large number of studies have demonstrated that the output response of a nonlinear system is generally influenced by the excitation level. For the PEHS, two interesting phenomena except the increase in the voltage output can be observed when the excitation amplitude is increased from 0.1 g to 0.35 g. First, the softening response is highlighted under higher excitation amplitudes, which can be identified through a comparison of Figs. 7–9. Moreover, the shift of the operating frequency band toward the left is more remarkable with an increase in the excitation level. It is worth mentioning that, for D = 15 mm, increasing the excitation level fails to enlarge the maximum attainable voltage because the beam’s deflection is limited by the value of d or xc. However, this design may find its application in preventing the beam from being overstrained and in the meanwhile achieving improved energy harvesting performance provided that d (or xc) is equal to the maximum allowable deflection of the beam, which can be realized by adjusting the positions of external magnets. Another interesting finding from Figs. 8 and 9 is that, among the three different PEHS configurations (D = 15 mm, 16 mm, and 17 mm) considered, the PEHS with D = 15 mm generates the highest voltage output under the smaller excitation level

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Fig. 7. Frequency response of open-circuit voltage under various configurations (a = 0.1 g).

(a = 0.15 g) but produces the lowest voltage output under the larger excitation level (a = 0.35 g), implying that the optimal configuration may exist for different excitation levels. To have a complete picture of the voltage output of the PEHS under various excitation levels, further frequency sweeps are conducted by simulation to obtain the maximum voltage output from the PEHS with different values of D under various excitation amplitudes, as shown in Fig. 10, in which the peak voltage outputs of the PEHS and its linear counterpart under the manual sweep are also depicted for comparison. The manual sweep herein means that the dynamics of the system starts from the initial stationary state (x = 0, x_ = 0, V = 0) for each excitation frequency [50]. It is evident that the maximum voltage outputs produced by the PEHS under various frequency sweep operations exhibit the similar trend as a function of the excitation amplitude and mass-magnet gap. For the linear configuration, it should be mentioned that no stoppers are required and then the peak voltage is subject to the excitation amplitude. However, for demonstrating a visual comparison, the stoppers are still employed in the simulations and the results indicate that, except at the smallest value of D (14.5 mm), the vibration of the beam is not influenced by the stoppers. As shown in Fig. 10, the PEHS can produce higher peak voltage output than its linear counterpart under various frequency sweep operations except when D = 14.5 mm, which is the smallest mass-magnet gap for achieving a monostable configuration for the magnetization and dimensions of the magnets and the stiffness of the beam given this study. With an increase in the value of D, the magnetic coupling between the beam and the external magnets is weakened, and then the voltage output of the PEHS approaches that of the linear configuration. For the PEHS, the results plotted in Fig. 10 indicate that, if a  0.35 g, the maximum voltage output can be acquired when D = 14.5 mm, whereas the PEHS with D = 16.5 mm produces the highest voltage output when the excitation amplitude a varies between 0.35 g and 0.45 g (0.45 g is the largest excitation amplitude used in the simulations). As discussed above, a smaller value of D leads to a stronger magnetic coupling between the beam and external magnets, which contributes to a larger voltage output under the lower excitation level. However, the decreased value of d or xc caused by the smaller D limits the PEHS voltage output particularly under larger excitation levels. On the contrary, a larger D allows the beam to deflect within an increased range, and then enables the PEHS to deliver a higher voltage output under larger excitation levels. It can be

Fig. 8. Frequency response of open-circuit voltage under various configurations (a = 0.15 g).

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Fig. 9. Frequency response of open-circuit voltage under various configurations (a = 0.35 g).

expected that a larger D ( 16.5 mm) is required for maximizing the voltage output of the PEHS under excitation levels >0.45 g. Therefore, for low excitation levels, a small value of D is required for achieving a better performance; otherwise a large D is more suitable. Output power is one of the major concerns when we evaluate the performance of an energy harvester [58]. An in-depth analysis regarding the output power of an electromagnetic energy harvester is given by Stephen [58]. For the PEHS, we note that the frequencies corresponding to the maximum power output vary with the value of D. Therefore, to show a fair comparison, excitations with different frequencies but a constant amplitude of 0.15 g are used to obtain the optimal or near optimal power output from the PEHS with different values of D. It is well evident in Fig. 11(a) that the PEHS of different configurations can generate a larger power output than the linear PEH. However, the resistance required for the PEHS to deliver the optimal power increases with a decrease in the value of D, which is caused by the reduction of the frequency corresponding to the maximum power output as the optimal resistance is inversely proportional to the resonance frequency [59]. On decreasing the value of D, the magnetic coupling is strengthened, which brings about an increase in the power output.

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Fig. 10. Maximum open-circuit voltage outputs under various configurations and excitation amplitudes from: (a) upward sweep, (b) downward sweep, (c) manual sweep, and (d) linear configuration.

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Fig. 11. Power output of various PEH configurations at an excitation level of 0.15 g: (a) power versus resistive load and (b) power versus frequency.

With the optimal or near optimal resistance shown in Fig. 11(a), frequency responses of the PEHS with different configurations and of its linear counterpart can be determined, as shown in Fig. 11(b). Similar to the voltage responses shown in Figs. 7–9, the shift of the working frequency range toward the left is also revealed by the frequency responses of the PEHS power output. In addition, with decreasing the value of D, the softening response becomes more pronounced due to the increasingly enhanced magnetic coupling between the beam and external magnets. However, the softening hysteresis shown in Fig. 8 with D = 15 mm and a = 0.15 g is not noticeable in Fig. 11 (b). This is attributed to the extraction of energy from the vibrating beam for powering the resistive load, which reduces the vibration amplitude of the PEHS and then weakens the influence of the magnetic coupling introduced, obscuring the softening hysteresis. To gain a more comprehensive understanding regarding the effect of D on the PEHS power output at various excitation levels, both upward and downward frequency sweeps are performed to determine the power delivered to a constant resistive load of 1 MX. Although the optimal power output cannot be warranted with this load for all excitation levels and values of D, the near optimal power output achieved with this load can exhibit the general trend of the power output. In terms of the maximum power output, the PEHS with various configurations generates similar results under upward, downward, and manual sweeps, as shown in Fig. 12. The maximum power output here means the peak power obtained using the frequency sweep operation for a given excitation level and value of D. Similar to the observations in Fig. 10, the PEHS can produce larger peak power outputs than its linear counterpart under various frequency sweep operations except when D = 14.5 mm. With increasing the value of D, the PEHS power output reduces gradually to the level generated by the linear counterpart. For the PEHS, it can be observed that, for a small excitation level (< 0.05 g), the power output increases monotonously with decreasing the value of D. However, for a large excitation (a  0.05 g), the power output increases with a decrease in the value of D, and then decreases with decreasing the value of D further. As a result, for a harmonic excitation, different values of D may lead to different power outputs from the PEHS. However, the PEHS with D = 16.5 mm could guarantee the optimal or near optimal performance in terms of power output under the harmonic excitations considered in this study, as shown in Fig. 12.

3.3. Output response under random excitations The majority of ambient vibrations appear in random patterns with energy distributed mostly in low-frequency range. To characterize the behavior of the PEHS in real-world situations, we also subject it to low-frequency (< 50 Hz) Gaussian white noise excitation with zero mean and specific standard deviation r. In reality, the excitation will be neither purely harmonic nor completely random but, instead, somewhere between these two extremes. However, these two extremes provide a guide on how the PEHS will behavior in reality. Fig. 13 shows the open-circuit voltage waveforms generated by the harvesters of various configurations when r = 0.45 g. The voltage response of the PEHS is observed to be always larger than that of the linear configuration. Moreover, the voltage response of the PEHS is increased significantly when the mass-magnet gap D is decreased from 17 mm to 15 mm. However, due to the presence of the stoppers, most of the response peaks are limited if D = 15 mm. For a larger value of D, the beam can oscillate freely without touching the external magnets and then the voltage response is not influenced by the stoppers. To examine the operating spectra of various PEH configurations, fast Fourier transform (FFT) is conducted on the voltage signal shown in Fig. 13, and the results are plotted in Fig. 14. It is obvious that the attractive arrangement between the beam and external magnets makes the working frequency range of the PEHS shift to the left compared to the linear PEH. Owing to the softening behavior, the PEHS has much higher amplitude and larger bandwidth than the linear PEH. The smaller value of D, the higher voltage amplitude of the PEHS. This is a tenuous conclusion based on the results shown in Figs. 13 and 14 because the response of a nonlinear system is always dependent on the strength of the applied excitation.

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Fig. 12. Power output under various configurations and excitation amplitudes from: (a) upward sweep, (b) downward sweep, (c) manual sweep, and (d) linear configuration.

Fig. 13. Open-circuit voltage waveform of various PEH configurations under random excitation with r = 0.45 g.

Fig. 14. Spectra (FFT) of open-circuit voltage of various PEH configurations under random excitation with r = 0.45 g.

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Fig. 15. RMS voltage output from various PEH configurations under random excitations: (a) voltage output versus standard deviation and (b) voltage output versus standard deviation and mass-magnet gap.

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Fig. 16. Power output from various PEH configurations under random excitations: (a) power versus resistive load and (b) power versus standard deviation and mass-magnet gap.

Fig. 15 shows the voltage output of the PEHS with various configurations under Gaussian white noise with different strengths (r). It is evident that the PEHS delivers much larger voltage output than the linear PEH in terms of the rootmean-square (RMS) value when the standard deviation varies from 0.01 g to approximately 1 g. Among the three PEHS configurations, the PEHS with D = 15 mm produces the voltage that has the highest rate of rise with the excitation strength, but the rise of the voltage output slows down if r > 0.35 g. The reduced rate of rise in the voltage output is caused by the contact or collision between the beam and the stoppers, which appears more frequently if D = 15 mm and r > 0.35 g, leading to a slowed increase in the voltage output with the excitation strength. This observation is different from the case when a harmonic excitation is used, in which the voltage output from the PEHS is flattened if the beam is excited to touch the stoppers by overly large excitations, as shown in Fig. 9. Moreover, compared to the case using the harmonic excitation, the massmagnet gap has a similar effect on the PEHS voltage output under random excitations. That is the PEHS with a small value of D can generate a large voltage output for random excitations with very small strength, whereas a larger value of D is required for the PEHS to maximize the voltage output when the excitation is strong. Power outputs of various PEH configurations under the Gaussian white noise excitation are shown in Fig. 16 (a), in which the standard deviation r = 0.45 g. The PEHS of various configurations is observed to outperform the linear PEH in terms of the power output. Among the three different PEHS configurations considered, the PEHS with the smallest mass-magnet gap (D = 15 mm) generates the largest power output. However, the optimal resistive load for attaining the peak power output increases with a decrease in the value of D. This is engendered by the shift of the working spectra to the left with the reduced value of D since the optimal resistive load is inversely proportional to the resonance frequency [59]. Fig. 16(b) shows the maximum power output of the PEHS with various configurations when subjected to the Gaussian white noise with the standard deviation r varying from 0.01 g to 1.16 g. The maximum power output herein is obtained using the optimal resistive

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load for a specific D and r, meaning that different resistances are used to demonstrate the power output depicted in Fig. 16 (b). Similar to the observation shown in Fig. 15 (b), the mass-magnet gap is required to increase gradually with the enhanced excitation strength for generating the largest power output. In the case that the external excitations are unknown, the PEHS with D = 16.5 mm could guarantee the optimal or near optimal performance under both harmonic excitations and random excitations, as shown in Figs. 12 and 16, demonstrating its superior adaptation to the ambient excitations. 4. Conclusions In this paper, a theoretical model is established for predicting the dynamic response of the proposed PEHS. The model is experimentally validated through a series of frequency sweeps using the harmonic excitations, which involves both the case that the beam oscillates freely without touching the stoppers and the case that collisions between the beam and the stoppers occur. Then, the proposed PEHS is investigated numerically under both harmonic excitations and random excitations. Based on the results of numerical simulations, it can be stated that: 1. The attractive coupling between the beam and extern magnets enables the PEHS operating frequency range to shift to the lower frequency, which is conducive to the energy extraction from low-frequency excitations. 2. For a given harmonic excitation, the PEHS can generate larger voltage and power outputs than the linear PEH under both upward and downward frequency sweeps, making the PEHS particularly suitable for capturing energy from small excitations. 3. Under the Gaussian white noise with a specific strength, the PEHS can provide a larger power output than the linear PEH, but a larger resistive load is required for the PEHS to maximize the power output. 4. Although the optimal PEHS configuration in terms of the mass-magnet gap varies with the excitation level, there exists a certain gap with which the optimal or near optimal performance of the PEHS can be achieved under both harmonic excitations and random excitations, demonstrating the harvester’s predominant adaptation to the ambient excitations. 5. Declarations of interest None. Acknowledgements This research is supported by the National Natural Science Foundation of China (51777147) and the Natural Science Foundation of Shaanxi Province (2018JM5030). References [1] C. Wei, X. Jing, A comprehensive review on vibration energy harvesting: modelling and realization, Renew. Sust. Energ. Rev 74 (2017) 1–18. [2] X. Wang et al, Similarity and duality of electromagnetic and piezoelectric vibration energy harvesters, Mech. Syst. Signal Pr. 52–53 (2015) 672–684. [3] X. 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